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Deriving IS, LM and aggregate demand curves

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The 3 problems are attached in the file below. They are about long-run equilibrium values, short-run values, level of investment and interest rate, amongst other things.

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https://brainmass.com/economics/general-equilibrium/deriving-aggregate-demand-curves-565680

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1. Assume that the long-run aggregate supply curve is vertical at Y= 3,000 while the short-run aggregate supply curve is horizontal at P = 1.0. The aggregate demand curve is Y = 2(M/P) and M = 1,500.
a. If the economy is initially in long-run equilibrium, what are the values of P and Y?
Using the aggregate demand curve function of Y = 2(M/P) and the aggregate supply curve Y=3,000, I equaled both equations together to derive P* first.

Y = 2(M/P) and Y=3000
3000 = 2 (M/P)
3000 = 2 (1500/P)
P* = 1

Using P*= 1, I then calculated Y*.
Y = 2(M/P*)
Y = 2(1500/1)
Y = 3000

The long-run equilibrium values are P=1 and Y=3000 respectively.

b. What is the velocity of money in this case?
Velocity of Money is the ratio of a country's GNP, Y, to the total money supply, M.
V = GDP/M
V = 3000/1500
V = 2

The velocity of money is 2 in this case.

c. Suppose because banks start paying interest on checking accounts, the aggregate demand function shifts to Y = (1.5)(M/P). What are the short-run values of P and Y?

Using the revised aggregate demand curve function of Y = 1.5(M/P) and the short-run supply shown by P=1, which has already been provided, and I then calculated short run demand, the new value of Y.

Y = 1.5(M/P) and P=1
Y = 1.5 (1500/1)
Y* = 2250

Thus, the short-run values are Y*= 2250 and P*=1

d. What is the velocity of money in this case?
V = GDP/M
V = 2250/1500
V = 1.5

e. With the new aggregate demand function, once the economy adjusts to long-run equilibrium, what are P and Y?

With the Y = 1.5(M/P) function and the long-run supply function of Y=3000 value, I again equaled both equations to find P* first, and then Y*.

Y = 1.5(M/P) and Y=3000
3000 = 1.5 (1500/P)
3000= 2250/P
P* = 0.75

Using P*= 0.75, I then calculated Y*.
Y = 1.5(M/P*)
Y = 1.5(1500/0.75)
Y = 3000

This result is not a surprise, as the shift in demand to a new function has enabled a lower price, but no effect on the long-run supply, which remains Y=3000.

f. What is the velocity now?
V = GDP/M
V = 3000/1500
V = 2

2. Assume that an economy is described by the IS curve Y= 3,600 + 3G-2T -150r and the LM curve Y= 2(M/P) + 100r [or r= 0.01Y-0.02(M/P)]. The investment function for this economy is 1,000-50r. The consumption function is C = 200 + (2/3)(Y-T). Long-run equilibrium output for this economy is 4,000. The price level is 1.0.

a. Assume that government spending is fixed at 1,200. The government wants to achieve a level of investment equal to 900 and also achieve Y = 4,000. What level of r is needed for I= 900? What levels of T and M must be set to achieve the two goals? What will be the levels of private saving, public saving, and national saving? (Hint: Check C +I+ G = Y.)

Recall the I = 1000-50r equation to find r.
I = 1000-50r
900 = 1000-50r
50r = 1000-900
50r = 100
r = 2

Also, remember that the IS curve equation has T component still unanswered for; however, we know now what r, G, and Y will be, and these are the values of 2, ...

Solution Summary

These questions respond to a few basic questions concerning how to derive the mathematical equations for aggregate demand, the IS and LM curves, which are all supported by Excel charts. Additionally, changes in monetary and fiscal policy are examined and analyzed further upon the direction the IS and LM curves shift, and the implications for investment opportunities if there is a "tight fiscal, loose money" or a "loose fiscal, tight money" policy mix.

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